Mathematician here. I understand why some people interpreted this post as dismissive of the article's premise. But I wish it could be okay, on a forum that putatively exists for founders and makers and challengers of the status quo, to say "this doesn't make sense when I think about it on a deeper level". Even if the reason it doesn't make sense to the speaker is because the speaker isn't an expert, and ultimately the thing does make sense.
A challenge or contradiction doesn't have to be disparaging. We are all laymen sometimes, and sometimes we just want to scratch an itch. It usually comes from a place of curiosity, not arrogance. We want to understand how something that seems weird to us can actually make sense, even if our understanding of only limited and superficial.
Why not let people express that in a direct way, without a bunch of deferential throat-clearing ("not an expert, just my thoughts here, this doesn't make sense to me...")? Who is hurt?
Great comment, thank you. Also, bravely asking these obvious, "stupid", elephant-in-the-room, naive questions on HN is a great place for a discussion to start, and will attract people who can explain things in simple terms. The comments will be more useful reading to more people than ones on a high, technical level. 'Being wrong on the internet' attracts truth. It's not the experts who will object to such questions!
And as a complete non-expert in this subfield of math (or anything within a hundred miles of it) I would love to hear why people are saying stuff like "equality is a fundamental mistake of mathematics and this approach is fixing it" rather than "this approach is developing a more refined/general/flexible notion of equality".
After a few minutes of deferential throat clearing, I'd like to say "in all seriousness, come on, equality isn't a mistake, and whatever brilliant insight your theory affords still isn't going to do away with it."
Programs and logical proofs are "isomorphic" according to Curry-Howard. So logical theories like math in general are "programs". Two programs can be "equal" in the sense that same inputs produce the same outputs, yet their source-code can be very different.
Therefore to explain a theory to a layman or anybody consider that your theory is a program, and therefore there can be any number of equal/equivalent but different programs, i.e. representations of that theory that are "equivalent" but have a different representation.
Therefore, and this is the point I'm trying to make, (and maybe I'm wrong) any good explanation of a theory could also be presented in some other way, and possibly in a simpler way. Programmers know this intuitively: You can write the same program in a complicated way or in a much simpler way, sometimes. Therefore the fact that some math is difficult to understand may in fact not be the fault of the student, but of that "math" itself.
And therefore it is useful that people come up with questions about what they find difficult to understand about a given theory. The "stupid" questions reveal something about the theory, they reveal what is difficult about it, what kind of mistakes people make when trying to understand it.
A challenge or contradiction doesn't have to be disparaging. We are all laymen sometimes, and sometimes we just want to scratch an itch. It usually comes from a place of curiosity, not arrogance. We want to understand how something that seems weird to us can actually make sense, even if our understanding of only limited and superficial.
Why not let people express that in a direct way, without a bunch of deferential throat-clearing ("not an expert, just my thoughts here, this doesn't make sense to me...")? Who is hurt?